An Introduction to Algebraic Topology

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Springer Science & Business Media, Jul 22, 1998 - Mathematics - 438 pages
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There is a canard that every textbook of algebraic topology either ends with the definition of the Klein bottle or is a personal communication to J. H. C. Whitehead. Of course, this is false, as a glance at the books of Hilton and Wylie, Maunder, Munkres, and Schubert reveals. Still, the canard does reflect some truth. Too often one finds too much generality and too little attention to details. There are two types of obstacle for the student learning algebraic topology. The first is the formidable array of new techniques (e. g. , most students know very little homological algebra); the second obstacle is that the basic defini tions have been so abstracted that their geometric or analytic origins have been obscured. I have tried to overcome these barriers. In the first instance, new definitions are introduced only when needed (e. g. , homology with coeffi cients and cohomology are deferred until after the Eilenberg-Steenrod axioms have been verified for the three homology theories we treat-singular, sim plicial, and cellular). Moreover, many exercises are given to help the reader assimilate material. In the second instance, important definitions are often accompanied by an informal discussion describing their origins (e. g. , winding numbers are discussed before computing 1tl (Sl), Green's theorem occurs before defining homology, and differential forms appear before introducing cohomology). We assume that the reader has had a first course in point-set topology, but we do discuss quotient spaces, path connectedness, and function spaces.
 

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Contents

CHAPTER 0
1
Brouwer Fixed Point Theorem
2
Categories and Functors
6
CHAPTER 1
14
Convexity Contractibility and Cones
18
Paths and Path Connectedness
24
CHAPTER 2
31
Affine Maps
38
Homology and Attaching Cells
189
CW Complexes
196
Cellular Homology
212
EilenbergSteenrod Axioms
230
Chain Equivalences
233
Acyclic Models
237
Lefschetz Fixed Point Theorem
247
Tensor Products
253

CHAPTER 3
39
The Functor n₁
44
n₁Są
50
Free Abelian Groups
59
The Singular Complex and Homology Functors
62
Dimension Axiom and Compact Supports
68
The Homotopy Axiom
72
The Hurewicz Theorem
80
Exact Homology Sequences
93
Reduced Homology
102
Homology of Spheres and Some Applications
109
Barycentric Subdivision and the Proof of Excision
111
More Applications to Euclidean Space
119
Simplicial Approximation
136
Abstract Simplicial Complexes
140
Simplicial Homology
142
Comparison with Singular Homology
147
Calculations
155
Fundamental Groups of Polyhedra
164
The Seifertvan Kampen Theorem
173
Attaching Cells
184
Universal Coefficients
256
EilenbergZilber Theorem and the Kunneth Formula
265
Basic Properties
273
Covering Transformations
284
Existence
295
Orbit Spaces
306
Group Objects and Cogroup Objects
314
Loop Space and Suspension
323
Homotopy Groups
334
Exact Sequences
344
Fibrations
355
A Glimpse Ahead
368
Cohomology Groups
377
Universal Coefficients Theorems for Cohomology
383
Cohomology Rings
390
Computations and Applications
402
Bibliography
419
Notation
423
Index
425
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